General stability

The theory of the general stability is a branch of the Microéconomie. She seeks to explain how the level of Production is fixed and of Consommation of the goods and the Prix in a economy.

The theory of balance in the context of the other economic approaches

This article covers the neo-classic approach of the theory of general stability. All the not-neo-classic theses on the behavior of the markets do not appear in this article. In particular, see the articles dedicated for the approaches traditional, Marxist or institutionnalists for the establishment of the prix the Analysis input-output of Wassily Léontief and the Model of growth in linear Programming developed by John von Neumann.

The economists of the Austrian École consider that general stability is an imaginary construction which can be useful for the study of particular problems, but which does not describe any realizable situation. For them, the economy is in perpetual imbalance and the only subject of scientific study is that of the processes of evolution. They thus regard all the theory of general stability as without object, a way of seeking " under the lampadaire" (because one can make calculations), even if it is known well that it is elsewhere than his keys were lost.

General stability tries to give an including explanation of all the economy in a rising step (known as still Bottom-up or “of base”). By opposition, the Macroéconomie known as keynésienne adopts a downward step which share of more important aggregates. Since the bringing together by the microeconomic bases, the border tends to grow blurred. However, of many macroeconomic models consider only one “market of the goods” and observe for example which are its interactions with the Financial market. Generally, the models of general stability distinguish several markets from differentiated goods. The recent models are often very detailed and rest on data-processing bases.

History of the model of general stability

Leon Walras left the first neo-classic attempt model the prices for an economy as a whole. In its Elements of pure political economy or theory of the social richness (1874) Walras proposes a series of models, by adding each time an additional aspect (two types of goods, several types of goods, the production, the growth, the currency). Many readers of Walras called into question the relevance of its model because of the centralizing need for the introduction for a voluntary appraiser offer and require households to calculate the system of equilibrium price. However, this problem relates to less Walras than these successors since its step is not positive (to say " what est") but normative (to say " what owes être"). Nevertheless, Walras posed the bases of a research program largely followed by the economists of the XXe century aiming at modelling the operation of the savings known as in market (souvant called program " Hicks-Samuelson" , largely " positif" thus). Lastly, let us note that Walras was also interested in the conditions of unicity and of stability of balances.

Walras was also the first to have an intuition of the role of the groping to reach balance and in particular its impact on stability. The prices are announced with shouted and the agents indicate how much they wish to offer or ask each good. No transaction is treated as much as one is with imbalance. The positive prices of the goods in excess of request are re-examined with the hausse they are lowered for the goods with slack demand. All the question is to know under which condition this process will reach the balance in which the demands and the supplies balance -- except on the goods of null prices for which supply must exceed. Walras does not find an definitive answer with this question. And due: one century later, Sonnenschein will show that it is simply impossible to deduce the form from supply and demand of the economic agents only thanks to their behaviors maximisateurs (cf Théorème of Sonnenschein) and that consequently, in the general case, balance is neither single, nor stable. This theorem applies besides to all models where the agents behave in " takers of prix".

In the analysis of a partial balance, the price determination of a good is simple: it is considered that the price of all the other goods is constant. The Theory marshallienne of the offer and the request is an example of analysis in partial balance. The analysis in partial balance is not possible that if the effects of the first degree of a shift of the Courbe of request for example does not influence the Courbe of offer. The Anglo-Saxon economists were interested in general stability at the end of the years 1920 and 1930, when Piero Sraffa showed that the models marshalliens could not explain the convexity of the curve of offer for the consumer goods.

If an industry claims only little a Factor of production, a light increase in the production ( output ) in this branch will not influence with the rise the price of this factor ( input ). In first order approximation, the companies will not be able to establish of fall of the costs and the curve of offer of the branch will not go down.

If, on the contrary, an industry consumes much a Factor of production, an increase in its activity will appreciably raise the price of this supply and thus of its costs. But this factor is probably used in the production of substitutable goods with that produced by industry: if it increases, the prices of the subtituts should probably increase too. Consequently, the first order effects of a shift of the curve of offer for industry which we look at should also modify the curve of request for its production. General stability tries to explore the interactions between these markets.

The economists of continental Europe made important openings in the years 1930. The evidence that Walras gave existence of a general stability rested on a enumeration equations and variables. These demonstrations are not valid for non-linear systems of equations. They then do not guarantee that the prices are positive, the opposite would not have any direction in its representation. The replacement of certain equations by inequalities and the recourse to more rigorous mathematical assumptions made it possible to improve the model of general stability.

The contemporary interpretation of general stability in economy

The problem of Walras is simultaneously to determine the exchanged quantities and the prices making it possible to equalize supply and demand for these quantities. Formally, since the quantities offered depend positively on the prices and that the required quantities depend negatively on the prices, it shows that the problem can be written in the form of a system of simultaneous equations. Since for L goods, and thus L walked, one has 2L equations given by supply and demand and that there is 2L unknown (the L exchanged quantities and the L price), Walras from of deduced that this system, having as many equations as unknown factors, should have a solution.

However, it is rather well-known in mathematics that even an algebraic system of equations to two equations and two unknown factors can not have solutions or have an infinity of solutions. The problem of the existence of a solution to this problem (known as problem of general stability) will remain clear a long time unresolved in spite of notable efforts economists like Cassel or Wicksell. It will be necessary to wait until 1953 and the joint contributions of Kenneth Arrow and Gerard Debreu to put a term at this situation.

The contributions of Arrow and Debreu make much more than to solve the old problem of Walras, they inaugurate an entirely new manner to pose and solve theoretical problems in economy. It results from it a certain confusion between the theory from general stability in a strict sense and the development of the Microéconomie and the mathematical economy which it accompanies. To complicate the things, the approaches initially distinct from Arrow and Debreu, amalgamated then in common contributions, raise in complementary but rather different ways to think the economic theorization: more constructive and " intuitive" at Arrow, hypothético-more deductive at Debreu.

Two important points are to be noted here. On the one hand, formalization is mathematically " ensembliste" , she knows only whole of production and consumption and relations inside them (preferences for the whole of consumption and technologies for the whole of production). It is not any more refers to " functions of utilité" or with " functions of production". The approach used is not thus any more " marginaliste" (and even regarded at the time as " anti-marginaliste"), i.e., as in the neo-classic tradition, based on reasoning by small differences along graphs in functions. In addition, the approach is conceptually " conventionnelle". The subjacent epistemological idea is that it is impossible for us to represent us mathematically the economy " réelle" , but that it is always possible for us to have a logico-formal representation of what we think that the economy is. In other words, any definition of an economy is a " convention" abstracted that we are free to modify according to the questions to solve or our personal vision from what is an economy. On the other hand, once adopted such a convention, we must subject to the rules formal logic such that they apply in the axiomatic context of this convention to deduce some from the answers to the put questions. Any economic argument which does not specify as a preliminary the terms of convention in which it operates (its " hypothèses" of departure, but the term is unsuitable here) is a priori suspect to be fallacious, because unfalsifiable fault of a convention of reference.

The initial contribution of Debreu constitutes a " blow of force" in the economic thinking. It is one of the first times in the history of the economic thinking, which an author dares to begin his matter in defining mathematically an economy. This definition reduces the economy to a whole of lists: a list of goods, a list of consuming agents, a list of preferences on the goods associated with the list with the consumers, a list of technologies of production of the list of the goods. Technologies and preferences are supposed to check certain axiomatic, axiomatic corresponding to the minimal number of assumptions necessary to obtain the required result (they were weakened since). The consumers, like the producers using technologies, are posed " concurrentiels" with the direction where they consider the prices of the goods as data that they cannot handle by modifying their demands or their supplies. They also have complete information on the exchanged markets and goods, and those can the being without specific cost.

Three major theorems were shown within this framework. First of all, of the theorems of existence show that balances exist under certain abstract conditions. The first theorem, known as of the social wellbeing, states the conditions under which a balance is effective within the meaning of Pareto. The second theorem of the social wellbeing specifies that each optimum of Pareto corresponds to a system of price, still within a precise formal framework. All these results rest on a formalization using the Topologie, using concepts such as the separations of hyperplanes or the theorem of the point fixes.

Interpretations of these results are classified in three fields. First of all, if one considers goods divided into distinct geographical sites, the model of Arrow-Debreu is a space model, of international business for example.

Moreover, if one distinguishes the goods according to the date on which they are delivered; all the instantaneous markets are balanced at one moment taken for origin. The agents of the model have in the long term thus contracted of the deliveries. This intertemporal model provides that in the beginning, all the future markets are negotiated.

Finally let us imagine that the contracts specify in which state of nature the delivery is done: the cover of an insurance is exerted only in the event of disaster.

“Contract for the transfer has off has commodity now specify, in addition to its physical properties, its hiring and its date, year vent one the occurrence off which the transfer is conditional. This new definition off has commodity allows one to obtain has theory off free from any probability concept…” (Debreu 1959)

These interpretations can be combined. The model of Arrow-Debreu thus applies when the goods are indentifiés by the place and the date of their delivery, and under which conditions. It leads to a complete Système of price for contracts of the type: “A ton of gauged potatoes benottes, delivered to Geneva on June 5th, if it rained less 85 cm in Were worth since October 1st. ” A model of general stability with complete markets of this type is enough far from a realistic description of the business world.

The work of Arrow and Debreu knew a great repercussion in the community of the economists, not only because it brought a rigorous solution to the one of the oldest problems of the economic scene: famous the " invisible Hand " of Adam Smith, but especially because it made it possible to take again the research program Néo-classique with formal means much more powerful. The theory of general stability then will cause an immense whole of work, and one can speak " of age of or" of this theory for all the period going of the medium of the years 1950 with the beginning of the year 1970.

It would be fastideux to enumerate all the developments of them. Among most important, let us quote the extensions of the concept to dubious environments, with dynamic and multisector contexts, the establishment of a correspondence between general stability and the heart of a co-operative play (Scarf, 1966), the introduction of anticipations (concept of epsilon-balance of Radner).

This research program will enter gradually in crisis starting from the beginning of the year 1970, as it became obvious that it was not possible, in the context of this theory, to explain how an economy, initially out-balance, could converge, by a succession of exchanges carried out out-balance, towards a balance. In other words, one was able to show the existence of a balance in a vast whole of situations but not to explain how an economy could reach it within a framework of free and decentralized exchanges (results in particular due to Mantel, Sonnenschein and Debreu, 1974).

Intuitively, one can explain it thus. The system of equilibrium price is the minimal quantity of necessary information to coordinate the economic agents on a balance. I.e. while announcing with the agents a vector of equilibrium price, they will coordinate each other spontaneously on the balance corresponding of supply and demand. But " minimal" does not want to say " suffisant" , and it is there the direction of the results of Mantel and alii. The problem of coordination is thus moved towards a problem " of information". But another of the weaknesses of this theory is precisely to suppose that the economic agents have complete information on the goods and the state of the exchanges.

This is why starting from the middle of the years 1970, the economists theorists any more will not seek to deepen, except with the margin, the theory of general stability in the context initially built by Arrow and Debreu. They will be directed in other directions, privileging the questions of information and incentives, thus seeking a form or another of " dépassement" theory of general stability, " dépassement" it appears today very difficult to predict nature, but let us recall that the problem of Walras waited 70 years before being solved…

Some mechanisms being able to lead to the balance

A world marked by the scarcity and populated rational individuals

The economists are interested only in the situations where the scarcity of the resources obliges the human ones to make choices. They must thus classify their choices by order preferably with an aim of satisfying their wishes as well as possible, under constraint not to use more than that they have. The neo-classic economic theory postulates that this classification is rational, even if subjacent rationality is not apparent with the observer. Each agent acts to maximize its utility, on the one hand by transforming its resources into products which it will sell (behavior of producer), on the other hand by transforming its resources into consumer goods (behavior of consumer). In this model, the prices entirely reflect the behaviors of all the agents.

they determine them: the price changes modify the preferences of the economic agents, and stimulate or discourage consumption and production. In general, the rise (respectively the fall) of the price cause a drop in (resp: to increase) consumption and increases (resp: lower) the production; however there are sometimes opposite effects (the rise of a good of first required -- bread or potato, for example -- can cause a drop in the purchasing power of the consumers sufficiently so that they consume more of this good, to the detriment of more luxurious goods -- pastry makings or meat, for example).
they undergo them: the variations preferably of the economic agents lead them to modify the prices to which they are close exchanging their goods. Thus, in general, when the consumers decrease their request or when the producers increase their production, the prices drop; and conversely

One sees thus that a modification of price leads in general to an opposite movement, by the means of the modifications of the preferences of consumption and production.

The pricing of the rate of balance per groping

At the beginning, there is no reason but the quantities requested (those which the consumers wish to buy for a level of price given) and the quantities offered (those which the producers wish to sell for a level of price given) are equal: the market is not with the balance.

The price must thus be adjusted (with the rise or the fall according to the situation) so that new supply and demand are established at a new price.

Let us take the example of a good produced in insufficient quantity: demand is higher than. Those which really make a point of getting it will state that they agree to pay a price higher than the initial price to have it. New producers will find then the market profitable, whereas in same time, unquestionable consuming will give up consuming a good become too expensive: supply increases and decreases. Walras explains why a fictitious agent, the “appraiser”, receives information on the quantities which the agents wish to exchange and returns to them of the prices which inform them about the scarcity of such or such good. By a process of “groping”, i.e. of tests and errors, this mechanism will make it possible to find the price which ensures the equality between supply and: it is the “equilibrium price”. The “quantities of balance” (supply and demand) are those which the agents express at this price.

In general, it is frequent that because of the response times of the economic agents and their lacks of information, one observes an equilibrium price not, but a more or less cyclic price around the theoretical equilibrium price. The behavior of the prices can even be perfectly chaotic, with phases of expansion or slip (when anticipations of the agents are divergent, and that they seek to be aligned on the others) and of the brutal shocks (often with the fall, but sometimes with the rise, when all the agents realign their anticipations). It is in particular the case for speculative goods or whose utility is very subjective (objet d'art, goods financial, etc). And this, even if the speculative phenomena empirically observed can be analyzed, for example like rational Bulles (according to the logic of the Beauty contest described by Keynes).

Multiple correlations

In a capitalist economy, the prices and the produced quantities of the goods are bound and correlated: change of the price of a good, bread for example, influence incomes of the bakers. If the bakers have specific tastes, i.e. they spend their money in a particular way, they buy more alarm clock-morning than the average, one finds this impact in the incomes of the clock and watch makers. The price of the bread will be certainly affected. To establish the equilibrium price of only one good requires in theory an analysis which takes account of the million other goods available.

DEBATEs around general stability

Two attitudes currently prevail: some try to amend the model of general stability to bring it closer to reality, others think that it should be given up with the profit of pragmatic constructions, inspired by observations of the real economy.

Proposed amendments with the theory

By contruisant models starting from work of Arrow and Debreu, research butted against some difficulties. The results of Sonnenschein, Mantel and Debreu stipulate that any restriction on the form of function of request is arbitrary. Some think that implies that the model of Arrow and Debreu is stripped of empirical Fondement. At all events, the balances defined by Arrow-Debreu could not be single, stable and given.

It was affirmed that a model of groping corresponds to a model of planned centralized economy in opposition to a decentralized market economy. He was proposed models of general stability with another type of processes, less convincing. In particular, the economists imagined models where the agents can negotiate prices apart from balance, these negations being able to affect balances towards which the economy tends. Let us announce the Processus of Hahn, Edgeworth and Fischer.

The intertemporal model of Arrow-Debreu in which all the anticipated markets exist at the initial moment, for goods to be delivered in the future, can be transformed into a sequential model of temporary balances. Each sequence would correspond to a balance on a specific market, and this constantly. Roy Radner found that so that a balance can exist in this type of model, the agents (consumers and producers) must have unlimited capacities of calculations.

Although the model of Arrow-Debreu is established with a arbitrary Numéraire, the model takes account of the Monnaie. Frank Hahn for example, sought to develop models of general stability in which the currency is a determining factor.

Criticisms of general stability

Some critical of general stability reproach the models on these models for being brought back to mathematical virtuosity, without bond with the real economy.

“There desirable are endeavors that now not for the most kind off economic contributions although they are just lime pit mathematical exercises, not only without any economic substance drank also without any mathematical been worth” (Nicholas Georgescu-Roegen 1979).

“One finds now certain participations which pass for the most brilliant contributions to the economy, whereas they are only mathematical exercises, not only deprived of economic substance, but also without mathematical value. ”

Georgescu-Roegen gives in example an article where one considers more agents than there is points on a line.

In general criticisms reproach the assumptions of the theory for being unrealistic. Let us notice inter alia, the perfect Rationalité of the individuals, in particular; a Complete information of the prices now and in the future; a common Knowledge of the nature of the goods; the absence of radical Uncertainty, in particular on the future and the guarantee of possibly contractual engagements; conditions of a Perfect competition.

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Let us notice that the first result of Arrow and Debreu synthesizes the main argument for a Dérégulation of an economy, which would lead to an Pareto-effective balance . On the other hand, the direct dependence of this balance to the initial Allocation of the resources returns in the defense of a policy of Redistribution.

The Theorem of Sonnenschein refutes the unicity of general stability

Quotations

the Theorem of Sonnenschein-Mantel-Debreu shows that general stability is ultimately only one empty and unusable construction. Claude Mouchot ( economic Methodology 1996).
Will be necessary it to wait until 2054 (80 years) to see the abandonment (Theory of general stability) Claude Mouchot ( economic Méthodologie 1996).

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